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D A Figure 4.1: State space rep res entation of a system. Since C adj(sI-A)B is a poly nomial matrix, obviously all poles of G(s)(i.e. the values of s for which G(s)=oo) have to be roots of the polynomial det(sI-A). The roots of det(sI-A) coincide with the eigenvalues of A. Hence, all poles of G(s) have to be eigenvalues of A. Th opposite needs not be the case al ways, since roots of det(sI - A)might be cancelled in(4.38) and consequently they will not appear as poles of G(s). This is the case, when the realiz ation (A, B, C, D)is uncontrollable, unob servable, or both. On the other hand, if the realiz (A, B, C, D)is both controllable and observable, the roots of det(sI- A)equals the poles of G(s and the pole polynomial p(s) will be given by ps=det(sI-A) 441) This means, that the dimension of A can not be smaller than the McMillan degree of G(s) Hence, a st ate sp ace realization which is both controllable and observable is called a minimal realization. These results can be summarized as the following theorem ThId li maldlaaaw K aaamranlavpacta Let G(s)be a tra fer matric with a minimal realization(A, B, C, D)and let p(s be the Smith-McMillan pole polynomial of G(s. Then dim a= degp(s (4.42) Hence, the McMillan degree of G(s equals the dimension cf a minimal realization. Moreot the eigenvalues f A equal the poles af G(s) Note, that if(A, B, C, D) is a non-minimal realization, then the poles of G(s) constitute a proper sub set of the eigenvalues of A aea Fdh rrtahrvelce dbaspranAauact The transformation of a state space description to a transfer function description is unique given by(4.38 ). In contrast, there are several ways in which a transfer function can be transformed into a st ate space description. A straightforward approach would be to derive separate st ate space descriptions for each column in G(s), i.e. for each input, and then collect these separate state space descriptions into an overall state space model. Let Gp) be the ith ch that G(s)=(Gn(s), Gt(s).,Ghs) O=tt C)ntr) "#' ; % ! ) ",2! ""#! && ) ! " ""(! '% # ( % # ( # $ ! ",2! 6 4 "",!